Help with a seemingly trivial proof - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-22T09:51:46Z http://mathoverflow.net/feeds/question/52465 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/52465/help-with-a-seemingly-trivial-proof Help with a seemingly trivial proof in70x 2011-01-19T03:18:57Z 2011-01-19T08:06:43Z <p>I have taken discrete logic/maths sophomore year of college <em>(CS major)</em>, since then I have transferred and they are making me take something very similar (Foundations of Computer Science). Today my teacher was just going through some of the basics which I know pretty well. She then proceeded to write a proposition to prove that I am seeming to struggle with (strange I was good at these in the past). She was not clear on what method she wanted us to use or anything really, the question is my interpretation of what I think she wants us to prove, her accent or writing isn't to helpful :-. The question is as follows: *Prove x is odd if and only if 8 divides \$x^2 - 1\$, in more formally: \$x\ is\ odd \iff 8|x^2-1\$. Right now I am using the idea that d|n then n = dk.</p> <p>So I basically started like this (Is contrapositive the way to go? modulus? -- she really didn't say much about these things it was the first day)</p> <p>\$8k = x^2 - 1\$</p> <p>\$8k = 4n^2 - 1\$</p> <p>\$(2n + 1)^2 - 1 = 8k\$</p> <p>\$4n^2 + 4n = 8k\$</p> <p>\$n^2 + n = 2k\$ (divided by four)</p> <p>\$n(n + 1) = 2k\$ (stuck)</p> <p>I am clearly rusty on this stuff, if someone could at least point me in the correct direction it would be grateful, or provide me with a very similar example proof.</p> <p><strong>Thanks!</strong></p>